Generator-offset property: Difference between revisions

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A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following properties:

# The scale is generated by two chains of stacked copies of a ''generator''.

# The scale is generated by two chains of stacked copies of an interval called ''generator''.

# The two chains are separated by an ''offset'' (the difference between the first note of the second chain and the first note of the first chain).

# The two chains are separated by a different interval called the ''offset'' (the difference between the first note of the second chain and the first note of the first chain).

# The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called ''alternants'') a1 and a2. Note that a1 does not need to [[subtend]], i.e. occur as, the same number of steps as a2.)

# The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step.

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The [[Zarlino]] (3L 2M 2S) JI scale is an example of a GO scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7.

== Mathematical definition ==

More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''GO''' if it satisfies the following properties:

@@ Line 1: / Line 1: @@
 A scale satisfies the '''generator-offset property''' (also '''GO''', '''alternating generator''' or '''AG''') if it satisfies the following properties:
-# The scale is generated by two chains of stacked copies of a ''generator''.
+# The scale is generated by two chains of stacked copies of an interval called ''generator''.
-# The two chains are separated by an ''offset'' (the difference between the first note of the second chain and the first note of the first chain).
+# The two chains are separated by a different interval called the ''offset'' (the difference between the first note of the second chain and the first note of the first chain).
 # The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called ''alternants'') a<sub>1</sub> and a<sub>2</sub>. Note that a<sub>1</sub> does not need to [[subtend]], i.e. occur as, the same number of steps as a<sub>2</sub>.)
 # The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step.
@@ Line 8: / Line 8: @@
 The [[Zarlino]] (3L 2M 2S) JI scale is an example of a GO scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7.
 == Mathematical definition ==
 More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''GO''' if it satisfies the following properties: